dft_tools/doc/Transport.rst

.. index:: Transport

.. _Transport:

Transport calculations
======================

.. index:: Theory

Formalism
---------
The optical conductivity and the Seebeck coefficient in the direction :math:`\alpha\beta` are defined as

.. math::

   \sigma_{\alpha\beta} = \beta e^{2} A_{0,\alpha\beta}  \ \ \  \text{and} \ \ \  S_{\alpha\beta} = -\frac{k_B}{|e|}\frac{A_{1,\alpha\beta}}{A_{0,\alpha\beta}} 

in which the kinetic coefficients :math:`A_{n,\alpha\beta}` are given by

.. math::
  
   A_{n,\alpha\beta} = N_{sp} \pi \hbar \int{d\omega \left(\beta\omega\right)^n f\left(\omega\right)f\left(-\omega\right)\Gamma_{\alpha\beta}\left(\omega\right)}

with the transport distribution

.. math::
  
   \Gamma_{\alpha\beta}\left(\omega\right) = \frac{1}{V} \sum_k Tr\left(v_{k,\alpha}A_{k}\left(\omega\right)v_{k,\beta}A_{k}\left(\omega\right)\right)

.. index:: Prerequisite

Prerequisites
-------------

.. note::
   Coming soon!


.. index:: Using the transport code.

Using the transport code
------------------------

.. note::
   Coming soon!


.. index:: References

References
----------